\(\int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx\) [440]

Optimal result

Integrand size = 19, antiderivative size = 134 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=\frac {6 d \sqrt {a+b x}}{b^2 \sqrt [4]{c+d x}}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}-\frac {6 \sqrt {d} \sqrt {b c-a d} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right |2\right )}{b^2 \sqrt [4]{c+d x}} \] Output:

6*d*(b*x+a)^(1/2)/b^2/(d*x+c)^(1/4)-2*(d*x+c)^(3/4)/b/(b*x+a)^(1/2)-6*d^(1 
/2)*(-a*d+b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/4)*EllipticE(sin(1/2*arctan 
(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b*c)^(1/2))),2^(1/2))/b^2/(d*x+c)^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=-\frac {2 (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \] Input:

Integrate[(c + d*x)^(3/4)/(a + b*x)^(3/2),x]
 

Output:

(-2*(c + d*x)^(3/4)*Hypergeometric2F1[-3/4, -1/2, 1/2, (d*(a + b*x))/(-(b* 
c) + a*d)])/(b*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(3/4))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {57, 73, 836, 765, 762, 1390, 1388, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {3 d \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}}dx}{2 b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {6 \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6 \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

Input:

Int[(c + d*x)^(3/4)/(a + b*x)^(3/2),x]
 

Output:

(-2*(c + d*x)^(3/4))/(b*Sqrt[a + b*x]) + (6*(((b*c - a*d)^(3/4)*Sqrt[1 - ( 
b*(c + d*x))/(b*c - a*d)]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c 
- a*d)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]) - ((b*c 
- a*d)^(3/4)*Sqrt[1 - (b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(b^(1/4) 
*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b* 
(c + d*x))/d])))/b
 

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [F]

\[\int \frac {\left (x d +c \right )^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {3}{2}}}d x\]

Input:

int((d*x+c)^(3/4)/(b*x+a)^(3/2),x)
 

Output:

int((d*x+c)^(3/4)/(b*x+a)^(3/2),x)
 

Fricas [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((d*x+c)^(3/4)/(b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x + a)*(d*x + c)^(3/4)/(b^2*x^2 + 2*a*b*x + a^2), x)
 

Sympy [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{4}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((d*x+c)**(3/4)/(b*x+a)**(3/2),x)
 

Output:

Integral((c + d*x)**(3/4)/(a + b*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((d*x+c)^(3/4)/(b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^(3/4)/(b*x + a)^(3/2), x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((d*x+c)^(3/4)/(b*x+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x + c)^(3/4)/(b*x + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/4}}{{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:

int((c + d*x)^(3/4)/(a + b*x)^(3/2),x)
 

Output:

int((c + d*x)^(3/4)/(a + b*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=\frac {4 \left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, c +9 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a^{3} d^{3}-15 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a^{2} b c \,d^{2}+9 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a^{2} b \,d^{3} x +6 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a \,b^{2} c^{2} d -15 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) a \,b^{2} c \,d^{2} x +6 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a \,b^{2} d^{2} x^{3}-2 b^{3} c d \,x^{3}+6 a^{2} b \,d^{2} x^{2}-a \,b^{2} c d \,x^{2}-2 b^{3} c^{2} x^{2}+3 a^{3} d^{2} x +4 a^{2} b c d x -4 a \,b^{2} c^{2} x +3 a^{3} c d -2 a^{2} b \,c^{2}}d x \right ) b^{3} c^{2} d x}{3 a b d x -2 b^{2} c x +3 a^{2} d -2 a b c} \] Input:

int((d*x+c)^(3/4)/(b*x+a)^(3/2),x)
 

Output:

(4*(c + d*x)**(3/4)*sqrt(a + b*x)*c + 9*int(((c + d*x)**(3/4)*sqrt(a + b*x 
)*x)/(3*a**3*c*d + 3*a**3*d**2*x - 2*a**2*b*c**2 + 4*a**2*b*c*d*x + 6*a**2 
*b*d**2*x**2 - 4*a*b**2*c**2*x - a*b**2*c*d*x**2 + 3*a*b**2*d**2*x**3 - 2* 
b**3*c**2*x**2 - 2*b**3*c*d*x**3),x)*a**3*d**3 - 15*int(((c + d*x)**(3/4)* 
sqrt(a + b*x)*x)/(3*a**3*c*d + 3*a**3*d**2*x - 2*a**2*b*c**2 + 4*a**2*b*c* 
d*x + 6*a**2*b*d**2*x**2 - 4*a*b**2*c**2*x - a*b**2*c*d*x**2 + 3*a*b**2*d* 
*2*x**3 - 2*b**3*c**2*x**2 - 2*b**3*c*d*x**3),x)*a**2*b*c*d**2 + 9*int(((c 
 + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**3*c*d + 3*a**3*d**2*x - 2*a**2*b*c** 
2 + 4*a**2*b*c*d*x + 6*a**2*b*d**2*x**2 - 4*a*b**2*c**2*x - a*b**2*c*d*x** 
2 + 3*a*b**2*d**2*x**3 - 2*b**3*c**2*x**2 - 2*b**3*c*d*x**3),x)*a**2*b*d** 
3*x + 6*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**3*c*d + 3*a**3*d**2*x 
 - 2*a**2*b*c**2 + 4*a**2*b*c*d*x + 6*a**2*b*d**2*x**2 - 4*a*b**2*c**2*x - 
 a*b**2*c*d*x**2 + 3*a*b**2*d**2*x**3 - 2*b**3*c**2*x**2 - 2*b**3*c*d*x**3 
),x)*a*b**2*c**2*d - 15*int(((c + d*x)**(3/4)*sqrt(a + b*x)*x)/(3*a**3*c*d 
 + 3*a**3*d**2*x - 2*a**2*b*c**2 + 4*a**2*b*c*d*x + 6*a**2*b*d**2*x**2 - 4 
*a*b**2*c**2*x - a*b**2*c*d*x**2 + 3*a*b**2*d**2*x**3 - 2*b**3*c**2*x**2 - 
 2*b**3*c*d*x**3),x)*a*b**2*c*d**2*x + 6*int(((c + d*x)**(3/4)*sqrt(a + b* 
x)*x)/(3*a**3*c*d + 3*a**3*d**2*x - 2*a**2*b*c**2 + 4*a**2*b*c*d*x + 6*a** 
2*b*d**2*x**2 - 4*a*b**2*c**2*x - a*b**2*c*d*x**2 + 3*a*b**2*d**2*x**3 - 2 
*b**3*c**2*x**2 - 2*b**3*c*d*x**3),x)*b**3*c**2*d*x)/(3*a**2*d - 2*a*b*...